According to the RAW in order to use a shield (6 lbs) and a dagger (light) weapon to attack in the same round, you would need the feat Dual Wielder.
The rules for two-weapon fighting (not the fighting style) on PHB page 195 stipulate that to get a second attack both weapons used must be light. In the weapon descriptions, the heaviest weapon that is classified as a light weapon weighs 3 lbs.
The Two-Weapon Fighting fighting style just allows you to add your stat bonus to your ability modifier to the damage from your second attack.
The Dual Wielder feat allows you to use weapons that are not light.
For clarification purposes, you must have the Dual Wielder feat to get both attacks when using the two-weapon fighting style if both weapons are not light melee weapons. (That is to say, if one or both of the weapons is not a light melee weapon then you absolutely need the Dual Wielder Feat to get the bonus attack.)
Further clarification, according to the section on two-weapon fighting (not the fighting style) in the PHB page 195 (emphasis mine):
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon you are holding in the other hand.
In other words, to get the off-hand bonus attack both the weapons you are holding must be light melee weapons if you are only using the two-weapon fighting style without the Dual Wielder feat. The feat allows you to get the bonus attack even if the weapons are not light melee weapons. So in the above question, a dagger is a light melee weapon but the improvised shield is not, therefore the Duel Wielder feat is needed to attack with both weapons in the same round.
The addition of the Two-Weapon Fighting fighting style would allow you to add you stat bonus to the damage of the second attack.
No
But the argument is not so straightforward.
The problem with this question is that the RAW are murky enough to allow for some leeway in interpretation.
Going by RAI, the intention was clearly to use Two-Weapon Fighting with one-handed weapons, and not game the system by attacking with a versatile weapon while holding it in two hands, arguing that it still counts as a one-handed weapon for the Dual Wielder feat.
The Rules
Dual Wielder
You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.
While this qualifies the longsword for two-weapon fighting, the Versatile property states that:
Versatile
This weapon can be used with one or two hands.
A damage value in parentheses appears with the property—the damage when the weapon is used with two hands to make a melee attack.
This explicitly states the need to use both hands to make a two-handed attack and use the bigger weapon dice, thus disqualifying this use case from the Dual Wielder requirement (one-handed melee weapons).
The Conclusion
While you can use a thrown handaxe and then attack with the longsword, you can only do so using the one-handed dice the Versatile weapon tag provides.
Best Answer
No, and Yes.
First of we need to consider this post by Jeremy Crawford:
See also: Shield and Improvised Weapon
It is not RAI to use a shield or other improvised weapon with the Dual Wielder feat. The rest that follows assumes you talked your DM into allowing that.
One Bonus Action
You can attack twice in your attack action, after level 5.
You, however, only get one bonus action to make one more attack if you attack with a light weapon you could attack (axe, axe, Shield), if your DM allows you to attack with the shield.
Relevant text from PHB 195:
So, you get a total of 3 attacks.
Rage Bonus Does Apply
Rage is a modifier added to the damage, so the following rule (also PHB195 would apply):
Rage, however, is not an ability modifier, and still would apply. You wouldn't, however, add your strength modifier without the feat -- but since we're talking about with the feat your good on this front.
+3 to AC — Um... Maybe?
I'd argue it is either a shield or a weapon depending on how you are using it. I, personally, don't think a shield grants any more armor because it is being used a weapon. That said, using the shield is already against RAI, but might be allowed, and if the DM allows that, you could push for the +3 -- but I think it is a stretch.